Representation theory in complex rank, II

We define and study representation categories based on Deligne categories Rep(GLt)Rep(GLt), Rep(Ot)Rep(Ot), Rep(Sp2t)Rep(Sp2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research.